a study on the holding capacity safety factors for torpedo...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 102618, 18 pagesdoi:10.1155/2012/102618

Research ArticleA Study on the Holding Capacity Safety Factors forTorpedo Anchors

Luı́s V. S. Sagrilo,1 José Renato M. de Sousa,1 Edison C. P. Lima,1Elisabeth C. Porto,2 and Jane V. V. Fernandes2

1 Laboratory of Analysis and Reliability of Offshore Structures, Civil Engineering Department,Centro de Tecnologia, COPPE/UFRJ, Cidade Universitária, Bloco I-2000, Sala I-116, Ilha do Fundão,21945-970 Rio de Janeiro, RJ, Brazil

2 PETROBRAS Research and Development Center (CENPES), Cidade Universitária, Quadra 7,Ilha do Fundão, 21949-900 Rio de Janeiro, RJ, Brazil

Correspondence should be addressed to José Renato M. de Sousa, [email protected]

Received 19 January 2012; Accepted 29 March 2012

Academic Editor: Ioannis K. Chatjigeorgiou

Copyright q 2012 Luı́s V. S. Sagrilo et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The use of powerful numerical tools based on the finite-element method has been improvingthe prediction of the holding capacity of fixed anchors employed by the offshore oil industry.One of the main achievements of these tools is the reduction of the uncertainty related to theholding capacity calculation of these anchors. Therefore, it is also possible to reduce the valuesof the associated design safety factors, which have been calibrated relying on models withhigher uncertainty, without impairing the original level of structural safety. This paper presentsa study on the calibration of reliability-based safety factors for the design of torpedo anchorsconsidering the statistical model uncertainty evaluated using results from experimental testsand their correspondent finite-element-based numerical predictions. Both working stress design�WSD� and load and resistance factors design �LRFD� design methodologies are investigated.Considering the WSD design methodology, the single safety is considerably lower than the valuetypically employed in the design of torpedo anchors. Moreover, a LRFD design code format fortorpedo anchors is more appropriate since it leads to designs having less-scattered safety levelsaround the target value.

1. Introduction

Torpedo anchors, as shown in Figure 1, are a cost-effective and practical solution foranchoring tautmooring lines of floating units as shown in Figure 2. This anchor has a “rocket”shape and is embedded into the seabed by free fall using its own weight as driven energy.The structure of a typical torpedo anchor usually comprises four different components �1, 2�:

2 Journal of Applied Mathematics

�a� �b�

Figure 1: Typical torpedo anchor with four flukes: �a� conical tip and �b� top with detail of the padeye.

Wire rope

Chain

Chain

Chain

Torpedo

Anchor

Polyester rope

Taut-leg radius = 1× WDW

ater

dep

th(W

D)

Conventional mooring radius = 3 × WD

Figure 2: Torpedo anchor for mooring lines of floating units.

a padeye that connects the first chain segment of the mooring line to the anchor; a ballastedshaft of carbon steel; four flukes, which are approximately rectangular �torpedo anchors withno flukes may also be employed�; a conical tip that is designed to help the embedment of theanchor. Typical torpedo anchors weight varies from 350 kN up to 980 kN.

These anchors have been first employed to hold flexible risers just after theirtouchdown points �TDPs� in order to avoid that the loads generated by the floating unitmovements were transferred to subsea equipments, such as wet Christmas trees �1�. In 2003a high holding capacity torpedo anchor was designed to sustain the loads imposed by themooring lines of a floating production, storage, and offloading �FPSO� �3� and, since then,various other floating production systems have been anchored using the torpedo concept�4�. Furthermore, recently, Henriques Jr. et al. �4� presented a new torpedo anchor with a totalweight of 1200 kN and larger flukes that will be employed tomoor FPSOs in ultradeepwaters.These characteristics, altogether, significantly increase its hold capacity when compared to thetraditional anchors with 980 kN in weight.

Journal of Applied Mathematics 3

Due to their “rocket” shape, the load position and inclination with respect to theirflukes, traditional methods, such as conventional beam-column analysis procedures �other-wise known as the P-Y method� or the simple formulae proposed by API guidelines �5� forpredicting the axial capacity of slender piles, often cannot be directly employed to estimatethe holding capacity of these anchors.

However, numerical procedures, based on the finite element �FE� method, haverecently been proposed by Brandão et al. �2� and de Sousa et al. �6��, and experimentaltests performed by PETROBRAS have demonstrated their good accuracy �7�. An importantoutcome of these FE-based models is the lower uncertainty in their predictions whencompared to traditional methods. Therefore, it is possible to reduce the design safety factorsvalues that have been calibrated relying on prediction models with higher uncertainty,without impairing the original level of structural safety.

There is no specific guideline for the design of torpedo anchors. Therefore, it iscommon practice to follow the indications from API RP 2A �5� for piles foundations tosupport offshore structures. This guideline prescribes that the safety factor associated withthese piles, according to a WSD design methodology, under operating environmentalconditions and during producing operations is equal to 2.0. It seems that this relativelylarge safety factor is mainly related to the high prediction model uncertainty due to thelarge number of pile capacity prediction models available at the moment of the calibrationprocess.

This paper presents a study on the calibration of safety factors for the ultimate limitstate design of torpedo anchors. One important aspect of this study is the availability ofresults from holding capacity tests of six torpedo anchors installed in Campos Basin, offshoreBrazil �7�. These results made it possible to assess the model uncertainty statistics associatedwith the FE-based model proposed by de Sousa et al. �6�.

Both working stress design �WSD� and load and resistance factors design �LRFD�methodologies are investigated. Concerning the traditional WSD methodology, it is shownthat, for the same actual design safety level of the traditional offshore piles, its single safetyfactor can be significantly lowered. However, the use of WSD design methodology resultsin designs with very scattered safety levels depending on the ratio of the functional andenvironmental load actions. Aiming at overcoming this drawback, a LRFD methodologycalibration is also investigated in this paper. The results show that the structural safety levelsof LRFD-based designs are more uniform than the WSD-based ones.

In the following sections, firstly, a brief review of the FE model proposed by de Sousaet al. �6� is presented followed by a comparison between the theoretical and experimentalholding capacities. After that, the reliability based-design study is presented, and, finally, themain conclusions of this work are stated.

2. Finite-Element (FE) Model

2.1. Overview

Recently, de Sousa et al. �6� have proposed a FE model devoted to predict the undrainedload capacity of torpedo anchors. This model employs isoparametric solid finite elements tomodel both the soil medium and the anchor. These elements are capable of representing thephysical nonlinear behavior of the soil medium and of the anchor itself. Large deformationsmay also be considered. The soil-anchor interaction is ensured by surface-to-surface contactelements placed on the external surface of the anchor and the surrounding soil medium. A


general overview of this model is presented in Figure 3. Next, its main aspects are brieflyreviewed.

2.2. Soil Modeling

2.2.1. Constitutive Matrix

The soil is assumed to be a perfectly elastoplastic isotropic material with physical propertiesvariable with depth. Hence, the relation between stresses and strains is given in the followingform:

�Δσ� �[Bep] · �Δε�. �2.1�

The constitutive elastoplastic matrix, �Bep�, is given by �9� as follows:

[Bep]� �Bel� − �Bel� · {∂P��σ�, �m��/∂σ} · {∂F��σ�, �k��/∂σ}

T · �Bel�{∂F��σ�, �k��/∂σ}T · �Bel� · {∂P��σ�, �m��/∂σ}

. �2.2�

The elastic total stress constitutive matrix can be split into two parts, as follows �9�:

�Bel� � �Beff� ⌊Bpore

⌋. �2.3�

In order to represent the nonlinear material behavior of the soil, the Drucker-Pragermodel was chosen. This model approximates the irregular hexagon of the Mohr-Coulombfailure surface by a circle, and, consequently, the Drucker-Prager yield function is a cylindricalcone. The yield function, F�·, ·�, and plastic potential function, P�·, ·�, for this model are givenby �10� as follows:

F��σ�, �k�� √J2 k

�1�DP · I1 k

�2�DP,

P��σ�, �m�� √J2 m

�1�DP · I1 m

�2�DP,

�2.4�

where k�1�DP, k�2�DP, m

�1�DP, and m

�2�DP are stress state parameters. These parameters depend on the

adopted approximation to the Mohr-Coulomb hexagon and are functions of the internalfriction angle, φ, the dilation angle, ψ, and the cohesion, c, of the soil. Figure 4 presentsthree possible approximations. In the proposed model, the circumscribed Drucker-Prager ap-proximation is used. It is assumed that the Drucker-Prager circle touches the Mohr-Coulombhexagon at a Lode angle of 30◦, which leads to the parameters �9� as follows:

k�1�DP �

2 · sen(φ)√3 · [3 sen(φ)]

, m�1�DP �

2 · sen(ψ)√3 · [3 sen(ψ)]

, k�2�DP � m

�2�DP �

6 · c · cos(φ)√3 · [3 sen(φ)]

.

�2.5�

It is worth mentioning that the rules for designing deep offshore foundations, such asthe API RP 2A �5�, typically assume that loading occurs rapidly enough so that no drainage


�a�

Ha

Hp

He

20D

�b�

Figure 3: General view of a typical FE mesh.

σ3

σ2

σ1

CircumscribedDrucker-Prager(extension )

InscribedDrucker-Prager

CircumscribedDrucker-Prager(compression )

Mohr-Coulomb

Figure 4: Drucker-Prager and Mohr-Coulomb yield surfaces in the deviatoric plane.

and hence no dissipation of excess pore pressures occur. This leads to an undrained conditionand, as a consequence, the undrained shear strength of the soil, su, may replace the cohesion,c, in �2.5�.

On the other hand, if the rate of loading is slow enough such that no excess porepressures are developed and sufficient time has elapsed since any previous application ofstresses such that all excess pore pressures have been dissipated, a drained analysis wouldhave to be performed. Lieng et al. �11�, however, indicate that the undrained loading capacityof a deep penetrating anchor, such as a torpedo anchor, is quite close to the drained capacity,and this is also assumed in this work. Nevertheless, this aspect has yet to be validated fortorpedo anchors.


2.2.2. FE Mesh Characteristics

The undrained response of a torpedo anchor embedded in clay is a classical J2 elastoplasticproblem that requires no locking for incompressible materials and reasonable bendingbehavior in order to obtain acceptable answers. Moreover, as high plastic strains are expectedin the soil, the elements may become highly distorted, and insensitivity to these distortionsis demanded. Finally, coarse meshes are often required for computational efficiency, andelements capable of providing sufficient accuracy are also needed. Aiming at addressingall these requirements, a possible choice, according to Wriggers and Korelc �12�, is theuse of enhanced strain hexahedrical and prismatic isoparametric finite elements with threedegrees of freedom at each node: translations in directionsX, Y , and Z. These elements avoidboth bend and volumetric locking by adding 13 internal degrees of freedom to the element�13, 14�.

An outline of the main dimensions of the soil mesh is shown in Figure 3. The proposedmesh is a cylinder with a base diameter of 20D, where D is the pile diameter. The height ofthe cylinder is given by the sum of the penetration of the torpedo anchor, Hp, the length ofthe anchor,He, and the distance of the tip of the torpedo anchor to the bottom of the FEmesh,Ha.

Each “slice” of the cylinder has its own physical properties, and, consequently, variablestrength, density, longitudinal and transverse moduli, and Poisson coefficients may beassumed.

The vertical displacement of the cylinder is restrained at the nodes associated with itsbase. The displacements in X and Z directions �see Figure 3� of nodes in the outer wall of thecylinder are restrained. Once a plane of symmetry is found, the out-of-plane displacementsare also restrained. In order to avoid any influence of these boundary conditions on theresponse of the anchor, several mesh tests were performed. In these tests, the cylinderdiameter and the distance from the tip of the anchor to the bottom of the mesh �Ha� werevaried. It was observed that a diameter of 20D for the cylinder and a distance of 5.0m forHawas enough to simulate an “infinite” media.

2.3. Anchor Modeling

The torpedo anchor is modeled with eight node isoparametric solid elements analogousto those used in soil representation. These elements, as pointed out before, are capable ofconsidering both material and geometrical nonlinearities. A typical FE mesh of a torpedoanchor is shown in Figure 5.

In order to properly represent the bending of the flukes and the shaft and,consequently, avoid shear locking problems, the numerical integration of each element isperformed with the enhanced strain method.

It is worth mentioning that neither the padeye at the top of the anchor nor the mooringline is represented in the proposed model. The load from the mooring line is applied at thegravity center of the padeye, where a node is placed and rigidly connected to the top of theanchor by rigid bars, as presented in Figure 6.

2.4. Anchor-Soil Interaction

The model proposed by de Sousa et al. �6� employs contact elements that allow relative largedisplacement and separation between the surfaces in contact. These elements are placed onthe external surface of the torpedo anchor and on the surrounding soil. A contact detection


�a� �b�

Figure 5: General view of a FE mesh for a torpedo anchor: �a� top and �b� bottom part of the flukes.

Rigid bars

Node where the load is applied

R = L + E

Figure 6: Load application.

algorithm based on the pinball technique and contact forces evaluated with the augmentedLagrangian method are employed in the formulation of these elements.

Relative sliding between the anchor and the soil is allowed when the shear stressesalong the outer wall of the anchor exceed the values estimated with the Mohr-Coulombfriction model, which is given by �5� as follows:

τmax�z� � α�z� · su�z� K0�z� · po�z� · tan(χ). �2.6�

The friction angle between the soil and the anchor wall, χ, is stated as �5�

χ � φ − 5◦. �2.7�


The adhesion factor, α, may be calculated by the following formula �5�:

α�z� �

⎧⎨

⎩

0.5 · ψ�z�−0.5, ψ�z� ≤ 1.0,0.5 · ψ�z�−0.25, ψ�z� > 1.0,

�2.8�

where

ψ�z� �su�z�po�z�

. �2.9�

2.5. Solution Procedure and Implementation

The FE analysis is divided in three different steps. In the first one, the initial stress state,that is, the stresses in the soil prior to the imposition of any kind of structural load to theanchor, is generated. The model proposed by de Sousa et al. �6� does not simulate the anchorpenetration in the soil, and it is assumed, by hypothesis, that the anchor is loaded with thesoil stress state undisturbed by the installation process. The anchor is thereby supposed to be“wished in place,” and this undisturbed stress state is imposed to the soil in the first step ofthe analysis.

However, the undrained shear strength of the soil and its stress state are deeplyaffected by the dynamic installation process and the setup time for reconsolidation orrecovery of shear strength after installation �15�. The direct consideration of this modifiedstress state and undrained shear strength is quite complex and, therefore, the hypothesis ofan undisturbed stress state, as employed in this work, is commonly assumed in the designof different types of offshore anchors such as suction anchors or other types of dynamicpenetrating anchors �16–18�. However, the validity of this hypothesis is subject of intensedebate. Lieng et al. �11�, relying on FE simulations, postulate that reconsolidation afterinstallation of a typical dynamic anchor would occur in less than six months, and, as aconsequence, a holding capacity determined from undrained analyses with an adhesionfactor close to 1.0 is expected to be valid. Richardson et al. �19�, relying on centrifuge testsand theoretical models, indicate that 50% of the operational capacity of a dynamic anchor isattained 35 up to 350 days after its installation. According to the same authors, 90% of theoperational capacity is reached 2.4 up to 24 years after the installation of the anchor. Thesevalues are extremely dependent on the horizontal coefficient of consolidation, and, therefore,authors recommend that dynamic anchors should be installed in soils that consolidationproceeds quickly. Furthermore, Richardson et al. �19� also indicate that values higher than1.0 for the adhesion factor are reached in the long term leading to the conclusion that theundrained holding capacity of the anchor would increase with time, and this capacity wouldeven be higher than the one estimated with an undisturbed stress state. Mirza �20� state thatthe time required by offshore piles to reach their full-operational capacity varies between1 and 2 years, and Jeanjean �21� indicates that 90% of the operational capacity of suctionanchors installed in soft Gulf of Mexico clays is attained in 90 days or less.

A typical torpedo anchor is firstly loaded approximately three months after itsinstallation, and, therefore, a great part of its full operational capacity has already beenreached. Moreover, the initial imposed loads are normally much lower than its actual loadcapacity, as its design relies on the maximum load obtained in mooring analyses that consider


both intact and damaged �one mooring line broken� conditions in the mooring system andalso several environmental load cases. These load cases encompass, for instance, extremewaves and currents. Hence, as the objective of the proposed FE model is to evaluate theundrained operational holding capacity of the anchor, it seems to be reasonable to assumethat full reconsolidation is achieved before the maximum operational load is imposed to theanchor. Nevertheless, in order to improve the proposed design method, more experimentaland also theoretical studies on this aspect have to be performed.

In the second step of the analysis, the self weight of the anchor is imposed. The analysisis restarted with the stress state from the first step, and gravity acts on the anchor. Finally, inthe third step, the total load is applied with adaptable increments since, as the surroundingsoil progressively fails, the stability of the solution is affected, and lower load increments arerequired. Typically, this last load step starts with an initial load increment of 5% of the totalload applied, and as the analysis progresses, it can be reduced to 1% of the total load.

At each load step, convergence is achieved if the Euclidian norm �L2� of the residualforces and moments is less than 0.1% of the absolute value of the total forces and momentsapplied.

The model was implemented in a program called ESTACAS. This software generatesFE meshes to be analyzed with ANSYS �22� program, where the following finite elementsare employed: SOLID185 in order to model the soil and the torpedo anchor; CONTA174 andTARGE170 are used to simulate the contact between the soil and the anchor.

3. Comparison with Experimental Tests

Aiming at validating the torpedo anchor geotechnical and structural design methodology,PETROBRAS conducted six full-scale tests in Campos Basin, offshore Brazil �7�. These sixfull-scale tests were divided into two sets.

In the first set of tests, three torpedoes �type 1�, each one weighing 350 kN, wereinstalled at a location with a water depth of 500m. At this location, soil was characterizedas being pure clay after cone penetration tests �CPTs� had been performed. The torpedoesembedment depths varied from 8m up to 11m. Besides, the torpedoes were pulled outwith load inclinations of about 40◦ �7�. The second set of tests consisted of pulling out threetorpedoes �type 2� weighting 430 kN each. The water depth at this second location is 150m,and the soil comprises both clay and sand layers. The torpedoes final embedment depths, inthis case, varied from 6m to 8m, and the load was imposed with inclinations varying from50◦ to 67◦.

Table 1 presents the relation between the experimental measured pull-out loads andthose estimated with the model proposed by de Sousa et al. �6�.

Despite all uncertainties related to the model and the geotechnical parametersinvolved, Table 1 indicates that the pull-out loads estimated with the FE model agree quitewell to the measured loads. Hence, these results will serve as a basis to the reliability-baseddesign that is presented next.

4. Reliability-Based Design

Modern probabilistic design codes aim at designing new structures with a specified targetprobability of failure. This can be achieved by fully probabilistic or deterministic codes �23�.The latter is more common in engineering design. In these codes the target probability offailure is not stated but it is implicitly achieved bymeans of safety factors. These safety factors


Table 1: Relation between measured and numerical pull-out loads.

TorpedoExperimental/numerical ratio

T35 T43

1 1.04 1.112 0.84 1.083 0.85 1.04

are calibrated by using standard structural reliability methods �23, 24�, in which all randomvariables �and their uncertainties� are represented by means of probability distributions.

The two main deterministic design methodologies for ultimate limit state �ULS� arethe working stress design �WSD� and load and resistance factor design �LRFD�. Focusingspecifically on the case of torpedo anchors for mooring systems �see Figure 6� under theWSDmethodology, the design equation can be written as

RkSF

≥ Lk Ek, �4.1�

where the characteristic functional load effect, Lk, is associated with the initial mooringline pretension applied to the anchor, and the characteristic environmental load effect, Ek,corresponds to the total tension acting on the anchor, when environmental loads are applied,minus the functional one. According to this classification, one can observe that the functionalload effect has smaller uncertainties than the environmental one.

In the case of the LRFD methodology, the design equation can be written as

Rk ≥ γLLk γEEk, �4.2�

where γL and γE are the partial safety factors associated with functional and environmentalload effects, respectively.

The safety factors of �4.1� and �4.2� are obtained by means of a calibration process. Themain idea of the calibration process is to obtain safety factors that guarantee a certain targetfailure probability, pfT , for the design. The target probability of failure may be stated as

pfT � Φ(−βT

). �4.3�

The calibration process is undertaken by means of reliability analyses consideringvarious design cases with different Ek/Lk ratios and also different statistical descriptionsof loadings and anchor resistances that are expected to be found in practice. Then, the safetyfactors can be defined, for instance, for LRFD methodology �similar procedure applies to theWSD methodology� through an optimization process to solve the following problem:

min g(γL, γE

)�

1N

N∑

i�1

(βT − βi

(γL, γE

))2, �4.4�

where βi is the reliability index of a given torpedo anchor design case that can be obtainedby a reliability analysis method, for example, first-order reliability method �FORM�, second-order reliability method �SORM�, or Monte Carlo simulation �23, 24�. In order to pursue this


kind of analysis, firstly, the limit state function must be defined. For the present study, it canbe stated as

G�X� � R − L − E, �4.5�

where X is the vector containing the random variables associated with the torpedo anchorholding capacity �R�, functional load �L�, and environmental load �E�. The variable R can berepresented by

R � C · RM, �4.6�

where RM is the predicted holding capacity evaluated by a given methodology of analysis,and C is the modeling uncertainty that represents the statistical bias between the measuredand predicted pile-bearing capacity. Therefore, some experimental data is necessary tostatistically define this random variable. In fact, C takes into account all uncertaintiesregarding the soil parameters, model representation, and so forth.

In this work, all reliability analyses have been performed withMonte Carlo simulationapproach which is applied to solve the following multidimensional integral:

pf � Φ(−β) �

∫

ΩfX�x�dx, �4.7�

where Ω is the failure domain, that is, G�X� ≤ 0.0, and fX�x� is the joint probabilitydistribution of the random variables X for the case under consideration.

4.1. Safety Factor Calibration

4.1.1. Random Variables Modeling

Tests with six torpedo anchors installed in clay and clay-sand soils have recently beenperformed �7�. The measured torpedo anchor holding capacities has been divided by theircorresponding predicted numerical estimation, RM, according to the model presented before,to determine themodel uncertainty,C, statistics. The values obtained are presented in Table 1.The mean, μ, and standard deviation, σ, for C according to Table 1 data correspond to0.993 and 0.118, respectively. In order to properly take into account the limited numberof experiments �N � 6� in the reliability analysis, it is necessary to apply some statisticalmodeling technique for reduced sample data as explained in Ditlevsen and Madsen �25�.Assuming that the C variable has a previous lognormal distribution, as indicated in Fenton�8�, this variable should be modeled by the following Student’s t-distribution to account forthe numberN of experiments:

fC�x� �1ξx

√d

d 2Γ��d 1�/2�

Γ�d/2�√dπ

(

1 t�x�2

d

)−�1/2��d1�; d � n − 1, �4.8�


where

t�x� �ln�x� − λ

ξ

√d

d 2, n � 6,

ξ �

√√√√ln

[

1 (σ

μ

)2]

� 0.1183; λ � ln(μ) − 0.5 · ξ2 � −0.0137.

�4.9�

Then, considering Table 1 data, the mean and standard deviation of this Student’s t-distribution becomes

μC � 1.003, σC � 0.193. �4.10�

It is important to notice that, due to the limited data sample, σC is 64% higher than thesample standard deviation, σ. These values also show that the FE model for the torpedo-holding capacity has a lower uncertainty when compared, for instance, with predictionmethods reported in an API-supported study for axial bearing capacity of single piles inclays �8�. In this latter case, the mean and standard deviation for the modeling uncertaintyvariable C has been, respectively, 1.04 and 0.34.

In the reliability analysis of marine structures, the functional loading is usuallymodeled by a normal distribution with a coefficient of variation �CoV or δ� in the order of0.05–0.10. In this work, a CoV �δL� of 0.07 is assumed. The environmental load effect is usuallyrepresented by its annual extreme value distribution which, in most of the cases, correspondsto a Gumbel distribution. Considering the inherent random variability of the environmentalparameters related to waves, wind and current, and also modeling uncertainties in theestimation of this complex load effect �26�, it is reasonable to assume a CoV �δE� in the orderof 0.10–0.30 for this distribution.

4.1.2. Characteristic Design Parameters

The characteristic design parameters used in �4.1� and �4.2� must be clearly defined in anycalibration process or design code. Different values for these parameters are also related todifferent sets of safety factors.

In the present study, the characteristic value of the torpedo anchor-bearing capacity,Rk, is defined as that predicted by the FE-based model previously described �6�. Thecharacteristic value of the functional load effect, Lk, has been assumed as its estimated meanvalue, that is, Lk � μL. In the case of the environmental load effect, following the traditionaldesign practice in offshore engineering, its characteristic value has been defined as the mostprobable value of the 100-yr extreme distribution. This value corresponds to that related to99% fractile of the annual extreme cumulative distribution �27�.

In the calibration process, a broader range of location water depths for the floatingunits has been assumed by considering the ratio Lk/Ek to be between 1.0 and 5.5. The lowerratio is associated with deeper waters in which the functional load at the anchor can be of thesame order as the environmental one. The converse applies to the higher ratio. These ratiosaccount for actual floating systems installed in water depths ranging approximately from200m to 2500m �28�.


4.1.3. Target Safety Index

In order to obtain the target safety index βT for the calibration process, firstly, all casesinvestigated have been designed according to the API guidelines �5� considering an intactmooring system, that is, using the WSD methodology with a safety factor equals to 2.0.Secondly, the reliability index of each case has been evaluated considering the uncertaintymodeling mean and standard deviation parameters as 1.04 and 0.34, respectively, taken fromFenton �8�. Finally, the target reliability index has been taken as the average of all thesereliability indexes. Considering specifically cases for δE � 0.10, 0.125, . . . , 0.30, the targetreliability index, βT , has been found to be approximately 2.9. Figure 7 illustrates the resultsthat have been obtained. In summary, the target reliability index assumed in this work is moreor less the same that is implicitly assumed when a single pile in clay is designed to supportan axial loading.

4.1.4. WSD Calibration

The optimizedWSD safety factor, SF , for the holding capacity design of torpedo anchors �see�4.1��, considering the previously discussed FE model uncertainty, has been found to be 1.45.Figure 8 shows a comparison between the target reliability index and the reliability indexesof all design cases investigated considering the optimized safety factor. When compared tothe safety factor of 2.0, which is usually employed by the offshore oil industry, the modelinguncertainty reduction obtained with the FE model causes a significant reduction in the safetyfactor.

Comparing Figures 7 and 8, it is important to notice that a larger model uncertaintyalso results in more scattered reliability indexes among the design cases.

4.1.5. LRFD Calibration

As it is well known in the public literature �29�, the use of a single safety factor in theWSD methodology leads to more scattered designs concerning their safety levels. The LRFDmethodology, which attributes partial safety factors for different loading sources �see �4.2��,is an alternative to reduce this scatter.

In order to obtain less-scattered reliability indices for design cases considered inthe calibration process, two sets of safety factors �γL, γE� have been identified. The LRFDmethodology design check for the holding capacity of torpedo anchors will be read as

Rk ≥ max�T1k, T2k�, �4.11�

where

T1k � 1.35 · Lk 1.43 · Ek,T2k � 1.72 · Lk 1.27 · Ek.

�4.12�

The two sets of safety factors in �4.12�, �1.35, 1.43� and �1.72, 1.27�, have been obtainedby applying the optimization process �4.4� twice: firstly, for all design cases in the range1.0 ≤ Lk/Ek ≤ 2.0 and, after that, for the other design cases in the range 2.0 < Lk/Ek ≤ 5.0.

A comparison between the target reliability index and the reliability indexes of allcases investigated considering the LRFD design, �4.4�, is shown in Figure 9.


1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 51 2 3 4 5

Ek/Lk ratio

2.2

2.4

2.6

2.8

3

3.2

3.4

Rel

iabi

lity

ind

ex(β)

WSD methodology

Averageβ

= 0.10 = 0.15 = 0.20 = 0.25 = 0.30

Design cases ( /δC = 0.34)SF = 2

CoV E CoV E CoV E CoV E CoV E

Figure 7: Reliability indexes of designs considering WSD methodology with SF � 2.0 and δC � 0.34 �8�.

1 2 3 4 51 2 3 4 5 1 2 3 4 51 2 3 4 5 1 2 3 4 5

Ek/Lk ratio

2.2

2.4

2.6

2.8

3

3.2

3.4

Rel

iabi

lity

ind

ex(β)

WSD methodology

Averageβ

= 0.10 = 0.15 = 0.20 = 0.25 = 0.30CoV E CoV E CoV E CoV E CoV E

Design cases ( = 1.45/δC = 0.19)SF

Figure 8: Reliability indexes of calibrated WSD designs considering δC � 0.19 �experimental tests�.

For bothWSD and LRFD design methodologies, the average of safety index values hasbeen the target reliability index βT � 2.9; however, their coefficients of variation have been,respectively, 4.5% and 2.5%. This shows that LRFD methodology gives designs with safetylevels that are slightly less scattered around the target value established for the calibrationprocess.

5. Final Remarks

This paper has presented a reliability-based design study for torpedo anchors consideringthat their ultimate bearing capacity is predicted numerically by a nonlinear finite elementmodel that represents the soil medium and the anchor itself. By considering the resultsof six full-scale experimental pull out tests, it was observed that the numerical model ispractically unbiased and presents a relatively low level of statistical uncertainty. Using themodel uncertainty statistics in reliability-based calibration process, it was shown that thesame level of structural safety implied in the traditional design of offshore piles can be


1

3.2

1 2 3 4 5 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

Ek/Lk ratio

2.2

2.4

2.6

2.8

3

3.4

Rel

iabi

lity

ind

ex(β)

LRFD methodologyDesign cases (δC = 0.19)Targetβ

= 0.10 = 0.10 = 0.10 = 0.10 = 0.10CoV E CoV E CoV E CoV E CoV E

Figure 9: Reliability indexes of calibrated LRFD designs considering δC � 0.19 �experimental tests�.

achieved with the use of lower safety factors. For instance, considering the WSD designmethodology, the single safety factor has a significant reduction dropping from 2.0 to 1.45.

This calibration study has also shown that a LRFD design code format for torpedoanchors is the most appropriate one since it leads to designs having less-scattered safetylevels around the target value. In this case, different safety factors are proposed for theenvironmental and functional parcels of the total load. These factors are also lower than theone proposed by API RP 2A �5�.

An important point to be noticed is that it is very likely that the safety factors presentedin this paper can become even lower when more experimental tests are performed since agreat part of the model uncertainty considered in the present work comes from the verylimited number of experiments available. More experiments are thereby required in order toestimate these new safety factors.

Nomenclature

Bel: Purely elastic total stress constitutive matrixBeff: Effective stress constitutive matrixBep: Elastoplastic constitutive matrix of the soilBpore: Constitutive matrix of the fluidc: Cohesion of the soilC: Model uncertaintyD: Diameter of the torpedo anchor including its flukesE: Environmental load effect applied to the anchorF�·, ·�: Yield functionG�·�: Limit state functionHa: Distance of the tip of the torpedo anchor to the bottom of the FE meshHe: Length of the torpedo anchorHp: Penetration of the torpedo anchor into the soil �embedment depth�I1: First invariant of the stress tensorJ2: Second invariant of the stress tensork � �k�1�, k�2��: Stress state parameters related to the yield function


K0: Coefficient of lateral earth pressureL: Functional load effect applied to the anchorLRFD: Load and resistance factors designm � �m�1�, m�2��: Stress state parameters related to the plastic potential functionN: Total number of design cases considered in the calibration processP�·, ·�: Plastic potential functionpo: Effective overburden pressurepfT : Target probability of failureR: Holding capacity of the anchorsu: Undrained shear strength of the soilSF : Safety factorWSD: Working stress designX: Random variables vectorz: Depth below mudline.

Greek

Δε: Incremental total strain vectorΔσ: Incremental total stress vectorα: Adhesion factorβi�·, ·�: Reliability index of ith case designed using the current set of safety factors �·, ·�βT : Reliability index associated with pfTχ: Friction angle between the soil and the torpedo anchor wallδ: Coefficient of variationφ: Internal friction angle of the soilΦ�·�: Standard normal cumulative distributionγL, γE: Partial safety factors applied to functional and environmental load effects,

respectivelyμ: Meanσ: Stress state in the soilσ: Standard deviationσy: Equivalent failure stressτmax: Limiting shear stressψ: Dilation angle of the soil.

Subscripts

C: Model uncertaintyDP: Related to the Drucker-Prager criterionE: Environmental load effectk: Related to characteristic �or nominal� valuesL: Functional load effectM: Related to model predictionsT : Target value.

Acknowledgment

The authors would like to thank PETROBRAS Research Center for allowing them to publishthe present paper.


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